Match List-I with List-II. Thermodynamics JEE Main PYQ, Partial Derivatives [2025]

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Solution

Q.
Match List-I with List-II. [2025]

List-I (Partial Derivatives)

List-II (Thermodynamic Quantity)

A. ${(\frac{\partial G}{\partial T})}_{P}$ I. $C_{p} $

B. ${(\frac{\partial H}{\partial T})}_{P}$ II. − S

C. ${(\frac{\partial G}{\partial P})}_{T}$ III. Cv

D. ${(\frac{\partial U}{\partial T})}_{V}$ IV. V

Choose the correct answer from the options given below:

1 A-II, B-I, C-IV, D-III

2 A-II, B-III, C-I, D-IV

3 A-II, B-I, C-III, D-IV

4 A-I, B-II, C-IV, D-III

Ans.
(1)

$G = H - T S$

$d G = d H - T d S - S d T$

$d G = d (U + P V) - T d S - S d T$

$d G = d U + P d V + V d P - T d S - S d T$

$d G = q_{rev} + W_{rev} + P d V + V d P - T d S - S d T$

$d G = T d S - P d V + P d V + V d P - T d S - S d T$

$d G = V d P - S d T$

At constant pressure, $d P = 0$ , so ${[\frac{\partial G}{\partial T}]}_{P} = - S$

At constant temperature, $d T = 0$ , so ${[\frac{\partial G}{\partial P}]}_{T} = V$

(B) $H = U + P V$

$d H = d U + P d V + V d P$

$d H = q_{rev} + W_{rev} + P d V + V d P$

$d H = q_{rev} - P d V + P d V + V d P$

$d H = q_{rev} + V d P$

$d H = n C d T + V d P$

At constant pressure $C = C_{p}$ and $d P = 0$

$d H = n C_{p} d T$

Taking 1 mole of substance, $(n = 1)$

${[\frac{\partial H}{\partial T}]}_{P} = C_{p}$

(D) $d U = q_{rev} + W_{rev}$

$d U = q_{rev} - P d V$

$d U = n C d T - P d V$

At constant volume, $C = C_{v}$ and $d V = 0$

$d U = n C_{v} d T$

${[\frac{\partial U}{\partial T}]}_{V} = C_{v}$

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